Ooops! I wrote it wrong, here's the problem:
if x varies directly as y^2, inversely as z, and inversely as the square root of w, what happens to x when y is multiplied by 3, z is doubled, and w is quadrupled?
I would really appriciate it if someone could help me understand how to do this problem. Thank you!
x = C y^2 z^-1 w^(-1/2)
C is a constant. Increasing y by a factor of three causes x to increase by a factor of 3^2 = 9. Doubling z causes x to decrease by a factor of 2. Quadrupling w doubles its square root, so the decrease in x is a factor of 1/2. Multiply the factors together: 9 x (1/2) x (1/2) and see what you get for the overall factor that x is increased.
thanks so much!
To solve this problem, we can start by writing the relationship between x, y, z, and w. We are given that x varies directly with y^2, inversely with z, and inversely with the square root of w. This can be represented by the equation:
x = C * y^2 / (z * sqrt(w))
Here, C is a constant of proportionality.
We are asked to determine what happens to x when y is multiplied by 3, z is doubled, and w is quadrupled.
Let's consider each change individually:
1. When y is multiplied by 3, the new value of y becomes 3y. According to the relationship x = C * y^2 / (z * sqrt(w)), increasing y by a factor of 3 means y^2 will be increased by a factor of 3^2 = 9. Therefore, the new value of x becomes 9x.
2. When z is doubled, the new value of z becomes 2z. As x varies inversely with z, doubling z will cause x to be divided by 2. Therefore, the new value of x becomes (9x) / 2 = (9/2)x.
3. When w is quadrupled, the new value of w becomes 4w. As x varies inversely with the square root of w, quadrupling w will cause the square root of w to be doubled. Since the square root of w is in the denominator, doubling it will result in x being divided by 2. Therefore, the new value of x becomes (9/2)x / 2 = (9/4)x.
Combining all these changes together, the overall factor by which x changes is 9/4. Therefore, x is increased by a factor of 9/4 when y is multiplied by 3, z is doubled, and w is quadrupled.